18.337 Final Project Presentation Richard Baumer

  Goal/Problem Statement:   Determine the transition state energy

barrier (ETS) between two minimized atomic configurations, R0 and RF

  Solution – Nudged Elastic Band (NEB) Method (Ref. 1): 1.  Interpolate between R0 and RF with a

sequence of N system images Ri

2.  Connect each image with an elastic spring, yielding a force Fi

S:

3.  Compute the interatomic – V(Ri) – forces via a classical or quantum method, Fi

V:

4.  Minimize atomic configuration Ri by

where τ is the tangent to the image path

Project Overview

€

R 0

€

R F

E

€

χ€

ETS

€

F i

S = ki+1( R i+1 −

R i) − ki(

R i − R i−1)

€

F i

V = − ∇ V ( R i)

€

F i =

F i

V − ( F i

V ⋅ ˆ τ ) ˆ τ [ ] + ( F i

S ⋅ ˆ τ ) ˆ τ [ ]∴ F i =

F i

V ⊥

+ F i

S ||

Serial NEB Results   Model System

  Lennard-Jones interatomic potential

  Relaxed FCC LJ Solid – 1 vacancy

  Model Problem

Determine the vacancy migration energy in an FCC LJ solid using the NEB Method

  Results  

€

V(rij ) = 4εσrij

⎛

⎝ ⎜

⎞

⎠ ⎟

12

−σrij

⎛

⎝ ⎜

⎞

⎠ ⎟

6⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

€

ETS = 0.83Ecoh

Parallelization Method

if rank == 0 !"Read vars from script; !

MPI:Bcast(vars) !

F_Partition_Load(rank) !

while(e_diff < e_cut) !"for i=start_num:end_num!" "read(Ri-1, Ri, Ri+1) !" "min_NEB(Ri) !" "write(Rinew) !"end !"MPI::Allreduce(E_Ri) !"e_diff=e_curr-e_prev!

end !

Enforced Synchronization

of processors

Distribute load by:   Bin {inc = floor(num_files/num_proc)} to each processor   Distribute {extra = num_files – inc*num_proc} evenly to {num_proc – extra} processors

Parallel NEB Results   Dramatic Parallel Speed-Up

  For 1 processor per image, a factor of 16 (compared to serial case) decrease in program runtime

  Decreased Parallel Efficiency   Parallel speed ratio (efficiency) decreases

due to required synchronization of processors at every NEB iteration

Conclusions   Implemented a serial NEB Method   Vacancy migration energy barrier found for an LJ

system

  Implemented a parallel NEB Method   Marked decrease in execution time with increasing

processor number   Parallel efficiency decreases with increasing processor

number

  Future Work   Automatic load-balancing/work allocation   Application of NEB method to other systems

References   NEB Method

  Jónsson, H., Mills, G., and Jacobsen, K. W.. “Nudged elastic band method for finding minimum energy paths of transitions.” in Classical and Quantum Dynamics in Condensed Phase Simulations, edited by B. J. Berne, G. Ciccotti, and D. F. Coker (World Scientific, Singapore, 1998), p. 385.

  Henkelman, G., Jonsson, H. J. Chem. Phys. 113 (2000) 9978.

  Steepest Descent Energy Minimization

  Numerical Recipes (C++). 3rd Edition.